# research

**Numerical Solution of Eigenvalue Problems**

This research interest originates mainly from my PhD under the supervision of Prof Heinrich Voss at the Hamburg University of Technology

**Quantum Nanostructures**

The unique physical properties and the potential for applications in micro- and optoelectronic devices made nanostructures like quantum dots, wells etc a subject of intensive research. In particular the discrete spectrum of nanostructures is of interest for the design of e.g. lasers, optoelectronic or spinotronic devices. The results in my thesis contribute to a better understanding of various aspects of the modeling of quantum nanostructures such as the effect of wetting layers or array structures. In my finite element models I also included the effects of spin-orbit splitting and external magnetic fields and proved the variational characterization for all these models. On the basis of the variational properties I developed robust and efficient methods for the solution of eigenvalue problems arising in the different models.

**Interior Eigenvalue Computation**

In the second part of my thesis I developed efficient numerical methods for the computation of a large number of interior eigenvalues and the associated eigenvectors of symmetric nonlinear eigenvalue problems. Such computations are an integral part of a wide range of acoustic simulations. For instance, a gyroscopic eigenvalue problem (a special case of a quadratic eigenvalue problem) arises in the simulations of the noise of rotating tires. Here the eigenfrequencies of interest lie in the audible range of the human ear, i.e. in the middle of the spectrum. The discretised problem is large and sparse. Together with the lack of a Schur form for quadratic problems this becomes a major computational challenge. I devised iterative methods, which work on the original problem without linearizing it first and thus avoiding doubling the size of the problem. These methods are capable of computing a large number of eigenfrequencies in the middle of the spectrum. The complexity per eigenvalue is constant regardless of the location of the desired eigenvalues in the spectrum. I adapted the methods to also handle rational eigenvalue problems, which arise for example in fluid structure interaction.

**QARPACK Quadratic Arnoldi Package**

Matlab Toolbox for iterative solution of large sparse quadratic eigenvalue problems. See the codes section of the website for more details.

**Numerical Analysis of Diffuse Optical Tomography Problem**

We are working towards rigorous numerical analysis of both the forward and inverse problem in Diffuse Optical Tomography. While the deblurring/deconvolution problems have been extensively analyzed, analogous results for DOT are still outstanding and the character of the inverse problem is very different to the deconvolution problem. In particular we are interested in the analysis of the convergence of Krylov methods and their optimal preconditioning for both the forward and inverse problem in DOT at each step of the nonlinear solver.

** Inverse Problems in X-ray Tomography**

During my time as PDRA in Prof Bill Lionheart’s group at the University of Manchester I developed a strong interest in inverse problems and, in particular, in image reconstruction. This continues in new directions in my EPSRC Postdoctoral Fellowship.

** Rebinning Methods for X-ray Cone Beam Computed Tomography (CBCT)**

In cone beam systems an acceleration of the imaging speed is usually achieved by moving to wider cone angles, which corresponds to collecting more images per rotation. Rapiscan in their new system, Real Time Tomography (RTT), instead eliminated the speed limiting factor altogether, replacing the mechanically rotating gantry with a stationary ring of sources, which can by quickly switched on and off by the on board electronics and multiple stationary rings of detectors. To accommodate the stationary ring of sources in the design, it was necessary to divert from the 4th generation CT geometry. The resulting new geometry requires substantially different reconstruction algorithms than those devised for the standard cone beam CT. My major contribution to the Manchester project was the development of a new class of reconstruction methods, Multi-Sheet Surface Rebinning Methods (MSSR), which successfully combine analytical and numerical approaches to reconstruction taking the best of both worlds. The methods fulfill all requirements posed by real time reconstruction i.e. they use localised data and are highly parallelizable resulting in overall extremely efficient algorithms.

**Optimal Design for Tomosynthesis**

Tomosythesis affords the opportunity for multiple views X-ray imaging. The major difference to the regular Cone Beam CT is the limited number of views, which allows only for low depth resolution in the reconstructed image. The cost of equipment can be a limiting factor for deployment of advanced techniques for luggage security screening. Therefore cost effective systems like the On-Belt Tomosynthesis, developed by Caroline Reid and Robert Speller, Medical Physics, UCL, which can be easily integrated with the existing structures, are of great interest. Clinical applications of Tomosynthesis are in e.g. breast cancer screening, where Tomosynthesis can potentially replace the widely used mammography scanners e.g. Dexela tomosynthesis system. I am interested in optimal design of such systems, as well as development of tailored methods for image reconstruction.

**EPSRC Postdoctoral Fellowship “Image Reconstruction: the Sparse Way”**

The goal of my fellowship is the development of new acquisition and reconstruction schemes for faster and more robust imaging using substantially fewer measurements than necessary today. In recent years, inspired by the success of algorithms like those used in the modern image compression standards JPEG and JPEG-2000, the new field of Compressed Sensing emerged quickly raising interest in sparsity-enhanced imaging across mathematics, computer science and many other areas.

Compressed Sensing ideas have found numerous applications in image processing, compression and coding and in analog-to-digital signal conversion. On the other hand applications to image reconstruction from a reduced number of physical measurements have been fewer and so is in general the mathematical theory to support them. The potential benefits are huge as it can be seen on a very successful application of Compressed Sensing ideas to Magnetic Resonance Imaging (Lustig, Stanford), where dramatic scan time reduction has been achieved. Though, major obstacles are yet to be overcome, which result from the fact that the standard formulation of the image reconstruction problem yields a not necessarily underdetermined but an ill-posed system. The ill-posedness may be only mollified by the alteration of the experiment towards the non-redundant information acquisition. Careful analysis is necessary to identify the limiting factors for each application. Establishing a firm mathematical foundation, and development of efficient and robust algorithms for sparsity-enhanced image reconstruction will first enable these exciting ideas to break through to a broader range of real-life image reconstruction applications.